Theorem phop 27673

Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
phop.2	|- G = ( +v ` U )
phop.4	|- S = ( .sOLD ` U )
phop.6	|- N = ( normCV ` U )

Assertion

Ref	Expression
phop	|- ( U e. CPreHil OLD -> U = <. <. G , S >. , N >. )

Proof of Theorem phop

Step	Hyp	Ref	Expression
1		phrel 27670	. . 3 |- Rel CPreHil OLD
2		1st2nd 7214	. . 3 |- ( ( Rel CPreHil OLD /\ U e. CPreHil OLD ) -> U = <. ( 1st ` U ) , ( 2nd ` U ) >. )
3	1, 2	mpan 706	. 2 |- ( U e. CPreHil OLD -> U = <. ( 1st ` U ) , ( 2nd ` U ) >. )
4		phop.6	. . . . 5 |- N = ( normCV ` U )
5	4	nmcvfval 27462	. . . 4 |- N = ( 2nd ` U )
6	5	opeq2i 4406	. . 3 |- <. ( 1st ` U ) , N >. = <. ( 1st ` U ) , ( 2nd ` U ) >.
7		phnv 27669	. . . . 5 |- ( U e. CPreHil OLD -> U e. NrmCVec )
8		eqid 2622	. . . . . 6 |- ( 1st ` U ) = ( 1st ` U )
9	8	nvvc 27470	. . . . 5 |- ( U e. NrmCVec -> ( 1st ` U ) e. CVecOLD )
10		vcrel 27415	. . . . . . 7 |- Rel CVecOLD
11		1st2nd 7214	. . . . . . 7 |- ( ( Rel CVecOLD /\ ( 1st ` U ) e. CVecOLD ) -> ( 1st ` U ) = <. ( 1st ` ( 1st ` U ) ) , ( 2nd ` ( 1st ` U ) ) >. )
12	10, 11	mpan 706	. . . . . 6 |- ( ( 1st ` U ) e. CVecOLD -> ( 1st ` U ) = <. ( 1st ` ( 1st ` U ) ) , ( 2nd ` ( 1st ` U ) ) >. )
13		phop.2	. . . . . . . 8 |- G = ( +v ` U )
14	13	vafval 27458	. . . . . . 7 |- G = ( 1st ` ( 1st ` U ) )
15		phop.4	. . . . . . . 8 |- S = ( .sOLD ` U )
16	15	smfval 27460	. . . . . . 7 |- S = ( 2nd ` ( 1st ` U ) )
17	14, 16	opeq12i 4407	. . . . . 6 |- <. G , S >. = <. ( 1st ` ( 1st ` U ) ) , ( 2nd ` ( 1st ` U ) ) >.
18	12, 17	syl6eqr 2674	. . . . 5 |- ( ( 1st ` U ) e. CVecOLD -> ( 1st ` U ) = <. G , S >. )
19	7, 9, 18	3syl 18	. . . 4 |- ( U e. CPreHil OLD -> ( 1st ` U ) = <. G , S >. )
20	19	opeq1d 4408	. . 3 |- ( U e. CPreHil OLD -> <. ( 1st ` U ) , N >. = <. <. G , S >. , N >. )
21	6, 20	syl5eqr 2670	. 2 |- ( U e. CPreHil OLD -> <. ( 1st ` U ) , ( 2nd ` U ) >. = <. <. G , S >. , N >. )
22	3, 21	eqtrd 2656	1 |- ( U e. CPreHil OLD -> U = <. <. G , S >. , N >. )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   <.cop 4183   Rel wrel 5119   ` cfv 5888   1stc1st 7166   2ndc2nd 7167   CVecOLDcvc 27413   NrmCVeccnv 27439   +vcpv 27440   .sOLDcns 27442   norm_CVcnmcv 27445   CPreHil OLDccphlo 27667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-1st 7168  df-2nd 7169  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455  df-ph 27668

This theorem is referenced by:  phpar  27679